# uniform convergence by mn test

**श्रेणी के एकसमान अभिसरण के लिए एमएन परीक्षण (Uniform Convergence By mn test):**

**श्रेणी के एकसमान अभिसरण के लिए एमएन परीक्षण (Uniform Convergence By mn test):**

- श्रेणी के एकसमान अभिसरण के लिए एमएन परीक्षण (Uniform Convergence By mn test):

Uniform Convergence of Sequences:Let the sequence <f_{n}(x)> converge for every point x in I i.e.,the function f_{n} tends to limit as n\longrightarrow{\infin} for every x in I.This limit will be a function of x,say f.It follows from the definition of a limit that for every \epsilon there exits a positive integer m such that

|f_{n}(x)-f(x)|<\epsilon \text{ whenever }n\geq{m}

This integer m will depend upon x as well as \epsilon and so we can write it as m\left(x,\epsilon \right).Now if we keep \epsilon fixed and very x, then for a given point x in I there corresponds a value of m\left(x,\epsilon \right).Thus we shall find a set of values of m\left(x,\epsilon \right). This set may or may not have an upper bound.If this set has an upper bound say M, then for every point x in I,we get

|f_{n}(x)-f(x)|<\epsilon \text{ whenever }n\geq{m}

In this case the sequence <f_{n}> is said to converge uniformly to f on I. __आपको यह जानकारी रोचक व ज्ञानवर्धक लगे तो अपने मित्रों के साथ इस गणित के आर्टिकल को शेयर करें।यदि आप इस वेबसाइट पर पहली बार आए हैं तो वेबसाइट को फॉलो करें और ईमेल सब्सक्रिप्शन को भी फॉलो करें।जिससे नए आर्टिकल का नोटिफिकेशन आपको मिल सके । यदि आर्टिकल पसन्द आए तो अपने मित्रों के साथ शेयर और लाईक करें जिससे वे भी लाभ उठाए । आपकी कोई समस्या हो या कोई सुझाव देना चाहते हैं तो कमेंट करके बताएं।इस आर्टिकल को पूरा पढ़ें।__

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Uniform convergence

**श्रेणी के एकसमान अभिसरण के लिए एमएन परीक्षण (Uniform Convergence By mn test):**

**श्रेणी के एकसमान अभिसरण के लिए एमएन परीक्षण (Uniform Convergence By mn test):**

- M_{n}-test:Let <f_{n}> be a sequence of functions defined on an interval I.

Let \lim_{n\longrightarrow{\infin}}f_{n}(x)\text{ for all }x\in{I}

Set M_{n}=\text{ sup }\left\{|f_{n}-f(x)|:x\in{I}\right\}

Then <f_{n}> converges uniformly to f if and only if M_{n}\longrightarrow{0}\text{ as } n\rightarrow{\infin}

- उपर्युक्त आर्टिकल में श्रेणी के एकसमान अभिसरण के लिए एमएन परीक्षण (Uniform Convergence By mn test) के बारे में बताया गया है।

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